IndisputableMonolith.Verification.JlogNonnegCert
Verification module packaging a non-negativity certificate for the logarithmic form of the Recognition cost J. Anyone checking that the J-cost is a genuine defect (vanishing only at the identity) cites this. The module imports the Cost layer and exposes a single certificate object whose fields pin Jlog ≥ 0 by elementary real analysis.
claimA certificate that $J_{\log}(t) := \cosh(t) - 1$ (equivalently $J(e^t)$) satisfies $J_{\log}(t) \ge 0$ for all real $t$, with equality precisely at $t = 0$.
background
Recognition Science forces a unique symmetric cost $J$ on the positive reals by the Recognition Composition Law and boundary conditions (forcing step T5). One closed form is $J(x) = (x + x^{-1})/2 - 1$, which rewrites as $J(e^t) = \cosh(t) - 1$. The logarithmic presentation $J_{\log}$ is the natural coordinate for ladder and eight-tick arguments.
Non-negativity of $J$ (and of $J_{\log}$) is the statement that $J$ is a true defect: it vanishes only at the fixed point $x = 1$ (equivalently $t = 0$) and is strictly positive elsewhere. The Cost module supplies the definition of $J$ and basic identities; this verification module isolates the non-negativity claim in certificate form so downstream proofs can invoke it without re-deriving elementary inequalities on $\cosh$.
proof idea
Definition-and-certificate module rather than a deep proof development. It introduces a certificate structure whose inhabitants package the inequality $J_{\log}(t) \ge 0$ together with the equality case. The underlying real-analytic fact is the standard expansion $\cosh(t) - 1 = 2\sinh^2(t/2) \ge 0$, or equivalently the AM-GM bound $(x + 1/x)/2 \ge 1$ for $x > 0$. No multi-step tactic script is required beyond discharging the certificate fields from Mathlib's hyperbolic or elementary inequalities.
why it matters in Recognition Science
Non-negativity of $J$ is a standing hypothesis for every defect-distance, mass-ladder, and forcing-chain argument that treats $J$ as a Lyapunov or cost functional. By isolating a reusable certificate, the module keeps those downstream developments free of local analytic lemmas. It sits in the Verification domain as a small, auditable pin for the T5 uniqueness story: once $J$ is forced, one must still know it is a genuine non-negative cost. No direct used-by edges are recorded yet; the intended consumers are Cost identities, defectDist constructions, and any proof that quotes $J \ge 0$ or $J_{\log} \ge 0$.
scope and limits
- Does not prove uniqueness of J; that is the T5 forcing result.
- Does not derive the Recognition Composition Law or any functional equation for J.
- Does not address discrete ladder gaps, masses, or dimension forcing (T6–T8).
- Does not supply numerical bounds on alpha, G, or other RS constants.
- Does not claim strict convexity or higher-order expansion coefficients beyond non-negativity.